DTZRZF(1) LAPACK routine (version 3.2) DTZRZF(1)NAME
DTZRZF - reduces the M-by-N ( M<=N ) real upper trapezoidal matrix A to
upper triangular form by means of orthogonal transformations
SYNOPSIS
SUBROUTINE DTZRZF( M, N, A, LDA, TAU, WORK, LWORK, INFO )
INTEGER INFO, LDA, LWORK, M, N
DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
PURPOSE
DTZRZF reduces the M-by-N ( M<=N ) real upper trapezoidal matrix A to
upper triangular form by means of orthogonal transformations. The
upper trapezoidal matrix A is factored as
A = ( R 0 ) * Z,
where Z is an N-by-N orthogonal matrix and R is an M-by-M upper trian‐
gular matrix.
ARGUMENTS
M (input) INTEGER
The number of rows of the matrix A. M >= 0.
N (input) INTEGER
The number of columns of the matrix A. N >= M.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading M-by-N upper trapezoidal part of the
array A must contain the matrix to be factorized. On exit, the
leading M-by-M upper triangular part of A contains the upper
triangular matrix R, and elements M+1 to N of the first M rows
of A, with the array TAU, represent the orthogonal matrix Z as
a product of M elementary reflectors.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,M).
TAU (output) DOUBLE PRECISION array, dimension (M)
The scalar factors of the elementary reflectors.
WORK (workspace/output) DOUBLE PRECISION array, dimension
(MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input) INTEGER
The dimension of the array WORK. LWORK >= max(1,M). For opti‐
mum performance LWORK >= M*NB, where NB is the optimal block‐
size. If LWORK = -1, then a workspace query is assumed; the
routine only calculates the optimal size of the WORK array,
returns this value as the first entry of the WORK array, and no
error message related to LWORK is issued by XERBLA.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
FURTHER DETAILS
Based on contributions by
A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
The factorization is obtained by Householder's method. The kth trans‐
formation matrix, Z( k ), which is used to introduce zeros into the ( m
- k + 1 )th row of A, is given in the form
Z( k ) = ( I 0 ),
( 0 T( k ) )
where
T( k ) = I - tau*u( k )*u( k )', u( k ) = ( 1 ),
( 0 )
( z( k ) ) tau is a
scalar and z( k ) is an ( n - m ) element vector. tau and z( k ) are
chosen to annihilate the elements of the kth row of X.
The scalar tau is returned in the kth element of TAU and the vector u(
k ) in the kth row of A, such that the elements of z( k ) are in a( k,
m + 1 ), ..., a( k, n ). The elements of R are returned in the upper
triangular part of A.
Z is given by
Z = Z( 1 ) * Z( 2 ) * ... * Z( m ).
LAPACK routine (version 3.2) November 2008 DTZRZF(1)
